It is possible to get lucky and avoid the rainy season, even in middle of June, but maybe don't count on it. I haven't been to Kyushu in June specifically, but in Shizuoka prefecture a couple of times in June and got very lucky with no rain and around 25 degrees C. But since you are already aware of the rainy season in Japan means you are already better prepared than most!

I can also add that I know of a guy who was stationed in Fukuoka for work and he did day trips to Kagoshima, Kumamoto and so on in the weekends, so it is possible if you get up early and plan accordingly.